Does there exist a differentiable function $f:\mathbb R \to \mathbb R^2$ such that $f([0,1])$ has a non-empty interior ? I know that such $f$ doesn't exist if I also assume $f$ is $C^1$ . Please help . Thanks in advance
Does there exist a differentiable function $f:\mathbb R \to \mathbb R^2$ such that $f([0,1])$ has a non-empty interior?
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multivariable-calculus
derivatives
metric-spaces
complete-spaces
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1Look for Peano curves, or space-filling curves. – 2017-01-24
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0Look up "space-filling curve". – 2017-01-24
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0@DanielFischer : Sorry for the confusion , I originally meant differentiable function . Are there differentiable space-filling curves ? – 2017-01-24
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0No - This was [discussed on MathOverflow](http://mathoverflow.net/questions/201424/proof-that-no-differentiable-space-filling-curve-exists). – 2017-01-24
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0@MarkMcClure : okay , so $f(\mathbb R)$ has $2$-dimensional Lebesgue measure 0 .. so inparticular it cannot contain any non-empty open set – 2017-01-24